
\begin{section}{Complex Functions}

\begin{subsection}{Linear Transformations}

A {\bf complex-valued function} $f$ is a function which maps each complex number $z$ in its domain $D$ to one and only one complex number $w$. We write $w = f(z)$ and call $w$ the {\bf image} of $z$ under $f$. The set of all such images $\{ w = f(z) \colon z \in D \}$ is called the {\bf range} of $f$.

Metaphorically speaking -- the domain of a function is the set of all arrows in a quiver, the image is the target to shoot at, and the range is all of the points on the target which are hit by an arrow. 

More rigorous set-theoretic definitions of the domain, range and image of functions can be found in most introductory calculus textbooks -- so for the sake of brevity will not be elaborated on here.

Just as we split a complex number into its real and imaginary parts to make it wieldy, we want to do the same with a complex valued function. Let $f \colon \mathbb{C} \to \mathbb{C}$ such that for any $z \in \mathbb{C}, f(z) = w$ define a mapping on the complex plane. Then

\begin{equation}
w = f(z) = f(x + iy) = u(x,y) + iv(x,y).
\end{equation}

\begin{ex}
Consider the function $f(z) = z^{4}$, which can be rewritten in the form $f(z) = u(x,y) + iv(x,y)$. That is,
$$f(z) = (x + iy)^{4} = x^{4} + 4x^{3}iy + 6x^{2}(iy)^{2} + 4x(iy)^{3} + (iy)^{4} = (x^{4} - 6x^{2}y^{2}+ y^{4}) + i(4x^{3}y - 4xy^{3}) = u(x,y) + iv(x,y).$$
\end{ex}

We can also decompose a function in polar form into component real-valued functions. Let $z = re^{i\theta}$, then

\begin{equation}
f(z) = f(re^{i\theta}) = u(r,\theta) + iv(r, \theta).
\end{equation}

\begin{ex}
Consider the function $f(z) = z^{2}$, which can be rewritten in the form $f(z) = u(r,\theta) + iv(r, \theta)$. That is,
$$f(re^{i\theta}) = (re^{i\theta})^{2} = r^{2}e^{2i\theta} = r^{2}\cos(2\theta) + i\sin(2\theta) = u(r,\theta) + iv(r,\theta).$$
\end{ex}

A good place to start with studying transformations on the complex plane is by looking at the elementary translation, rotation and scaling transformations.

\begin{defn}
Let $B = \alpha + i\beta$ denote a fixed complex number, then the transformation $$w = T(z) = z + B = x + \alpha + i(y + \beta)$$ is a {\bf translation} which displaces $z$ through the vector $B$ to its new position $w = T(z)$.
\end{defn}

The inverse map $T^{-1}(w)$ is trivially determined by substituting the plus signs with minus signs.

\begin{defn}
Let $\phi$ be a fixed real number, then for $z = re^{i\theta}$, the transformation $$w = R(z) = ze^{i\theta} = re^{i\theta}e^{i\phi} = re^{i(\theta + \phi)}$$ is a {\bf rotation} which rotates $z$ about the origin through the angle $\phi$ to its new position $w = R(z)$.
\end{defn}

The inverse map $R^{-1}(w)$ can be derived by simply raising the exponential function to a negative power.

\begin{defn}
Let $K > 0$ be a fixed positive real number, then the transformation $w = S(z) = Kz = Kx + iKy$ is a {\bf scaling} of $z$ by a factor of $K$.
\end{defn}

The inverse function can be derived by replacing $K$ with its reciprocal $\frac{1}{K}$. Note that if $K > 1$ we have a magnification, if $K < 1$ we have a shrinking, and if $K = 1$ we simply have the identity map.

Composing these concepts together, we can define a more generalized mapping on the complex plane.

\begin{defn}
  Let $A = Ke^{i\phi}$ and $B = \alpha + i\beta$ where $K > 1$ is a positive real number, then a {\bf Linear Transformation} is a mapping of the general form $$ w = L(z) = Az + B.$$
\end{defn}

Now we turn our attention briefly to power mappings of the form $w = z^n$ and $w = z^{\frac{1}{n}}$.

\begin{defn}
The function $$g(w) = w^{\frac{1}{n}} = |w|^{\frac{1}{n}} e^{i\frac{{\rm Arg}(w)}{n}}$$ is called the {\bf principal $n$th root function} and is the inverse of $w = f(z) = z^n$.
\end{defn}

\begin{rmk}
The mapping $w = z^2$ doubles the argument and squares the radius. The mapping $z = w^{\frac{1}{2}}$ similarly has half the argument and the square root of the radius.
\end{rmk}

\end{subsection}

\begin{subsection}{Limits and Continuity of Complex Functions}

The definition of a limit of a complex function parallels the definition of a limit of a real valued function in $\mathbb{R}^2$.

\begin{defn}
The expression $\displaystyle \lim_{z \to z_0} f(z) = w_o$ means that for each real number $\epsilon > 0$, there exists a real number $\delta > 0$ such that $$ |f(z) - w_0| < \epsilon$$ whenever $$ 0 < | z - z_0 | < \delta.$$
\end{defn}

The following theorem makes this definiton much more useful.

\begin{thm}
Let $f(z) = u(x,y) + iv(x,y)$ be a complex function defined in some neighborhood of $z_0$, except perhaps at $z_0 = x_0 + iy_0$. Then $$\lim_{z \to z_0} = w_0 = u_0 + iv_0 \iff \lim_{(x,y) \to (x_0,y_0)} u(x,y) = u_0 \land \lim_{(x,y) \to (x_0,y_0)} v(x,y) = v_0.$$
\end{thm}

The definition for the continuity of a complex valued function is also similar to a real function of two variables.

\begin{defn}
Let $f(z)$ be a complex function defined for all values of $z$ in some neighborhood of $z_0$. Then $f$ is {\bf continuous} at $z_0$ if all three conditions hold
\begin{enumerate}
\item $\displaystyle \lim_{z \to z_0}$ exists,
\item $\displaystyle f(z_0)$ exists,
\item $\displaystyle \lim_{z \to z_0} f(z) = f(z_0)$.
\end{enumerate}
\end{defn}

Again, the theorem following the definition is more useful than the definition itself.

\begin{thm}
Let $f(z) = u(x,y) + iv(x,y)$ be defined in some neighborhood of $z_0$. Then $f$ is continuous at $z_0 = x_0 + iy_0$ if and only if $u$ and $v$ are continuous at $(x_0,y_0$.
\end{thm}

\begin{rmk}
It is not surprising but worth noting that the sum, difference, product, quotient and composition of continuous complex functions will also be continuous.
\end{rmk}

\end{subsection}

\begin{subsection}{Differentiable and Analytic Functions}

Continuing onward with adapting our knowledge of elementary calculus to the calculus of complex functions, we say that that complex function $f(z)$ is {\bf differentiable at} $z_0$ if the following limit exists.

\begin{equation}
f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0} = \lim{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z}.
\end{equation}

Note that $\Delta z = z - z_0$ is only coherent if the first limit actually exists.

\begin{defn}
A complex function $f$ is {\bf analytic} at the pont $z_0$, provided there is some $\epsilon > 0$ such that $f'(z)$ exists for all $z \in D_{\epsilon}(z_0)$.
\end{defn}

In other words, $f$ is differentiable at $z_0$ and at all points in the $\epsilon$-neighborhood of $z_0$.

\begin{defn}
A complex function $f$ is {\bf entire} if it is analytic on the whole complex plane.
\end{defn}

Continuity is a weaker condition than differentiability. This leads us the the next theorem.

\begin{thm}
If $f$ is differentiable at $z_0$, then $f$ is continuous at $z_0$.
\end{thm}

\begin{thm}
Suppose that $f(z) = f(x + iy) = u(x,y) + iv(x,y)$ is differentiable at $z_0 = x_0 + iy_0$. Then the partial derivatives of $u$ and $v$ exist at the pont $(x_0,y_0)$ and 
\begin{equation}
f'(z_0) = \frac{\partial u}{\partial x} (x_0,y_0) + i\frac{\partial v}{\partial x} (x_0,y_0)
\end{equation}
and
\begin{equation}
f'(z_0) = \frac{\partial v}{\partial y} (x_0,y_0) - i\frac{\partial u}{\partial y} (x_0,y_0).
\end{equation}

Equating the real and imaginary parts of the above equations gives the {\bf Cauchy-Riemann Equations}
\begin{equation}
\frac{\partial u}{\partial x} (x_0,y_0) =  \frac{\partial v}{\partial y} (x_0,y_0) \land \frac{\partial v}{\partial x} (x_0,y_0) = - \frac{\partial u}{\partial y} (x_0,y_0).
\end{equation}
\end{thm}

This gives a better condition for differentiability.

\begin{thm}
Let $f$ be a continuous function defined in some neighborhood of $z_0$. Then if all partial derivatives of $f$ exist and are continuous and the Cauchy-Riemann Equations hold at $z_0$, then $f$ is differentiable and the derivative can be computed.
\end{thm}


\end{subsection}


\end{section}
